In Exercises 19–32, find the (a) domain and (b) range.
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𝔂 = -2 + √1 - x

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Step 1: Identify the expression inside the square root, which is '1 - x'. The square root function is defined only for non-negative values, so set up the inequality 1 - x ≥ 0.

Step 2: Solve the inequality 1 - x ≥ 0 to find the domain of the function. Rearrange the inequality to find the values of x that satisfy it.

Step 3: The domain of the function is the set of x-values for which the expression inside the square root is non-negative. Express this domain in interval notation.

Step 4: To find the range, consider the values that the expression √(1 - x) can take. Since the square root function outputs non-negative values, determine the minimum and maximum values of √(1 - x).

Step 5: The range of the function is determined by the expression -2 + √(1 - x). Calculate the minimum and maximum values of this expression based on the domain found in Step 3, and express the range in interval notation.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Domain of a Function

The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. For the function y = -2 + √(1 - x), the expression under the square root must be non-negative, which imposes restrictions on x. Thus, determining the domain involves solving the inequality 1 - x ≥ 0.

Range of a Function

The range of a function is the set of all possible output values (y-values) that the function can produce. For the function y = -2 + √(1 - x), the square root function outputs non-negative values, which means the minimum value of y occurs when x is at its maximum in the domain. Analyzing the function helps identify the range based on the values y can take.

Square Root Function

The square root function, denoted as √x, is defined for non-negative values of x and produces non-negative outputs. In the context of the function y = -2 + √(1 - x), the square root affects both the domain and range, as it restricts x to values where 1 - x is non-negative, and it shifts the output down by 2, impacting the overall range.

In Exercises 19–32, find the (a) domain and (b) range.____𝔂 = -2 + √1 - x